1 research outputs found
On the explosion of the number of fragments in the simple exchangeable fragmentation-coalescence processes
We consider the exchangeable fragmentation-coagulation (EFC) processes, where
the coagulations are multiple and not simultaneous, as in a
-coalescent, and the fragmentations dislocate at finite rate an
individual block into sub-blocks of infinite size. Sufficient conditions are
found for the block-counting process to explode (i.e. to reach ) or not
and for infinity to be an exit boundary or an entrance boundary. In a case of
regularly varying fragmentation and coagulation mechanisms, we find regimes
where the boundary can be either an exit, an entrance or a regular
boundary. In the latter regular case, the EFC process leaves instantaneously
the set of partitions with an infinite number of blocks and returns to it
immediately. Proofs are based on a new sufficient condition of explosion for
positive continuous-time Markov chains, which is of independent interest.Comment: 33 pages. No major modification. A technical Lemma (Lemma 5.3) has
been added and is used in the following work arXiv:2012.0857